Here is an example of a Lissajous curve tracer implemented in JavaSketchpad. To generate a Lissajous curve we want the two coordinates parametrized by sinusoidal curves of possibly different periods. It is easy to translate circular motion into sinusoidal curves by dropping a perpendicular. This version lets you keep track of the coordinates separately or together, adjust the periods by changing the diameters of the circles, and include guide lines to illustrate the horizontal and vertical components of the traced points.

The Sketchpad file can be downloaded if you would like to inspect or modify it. I find the trace prettier to watch on the applet version than on the Sketchpad version -- the shapes of the trace points provide it with a three dimensional quality.